What is the solution for 9y^{''}-y=xe^{x/3},y(0)=1,y^'(0)=0 ?

The solution for 9y^{''}-y=xe^{x/3},y(0)=1,y^'(0)=0 is

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9y^{''}-y=xe^{x/3},y(0)=1,y^'(0)=0

9 y^{^{''}} - y = x e^{\frac{x}{3}}, y (0) = 1, y^{^{'}} (0) = 0

y = (- \frac{1}{8} + \frac{5}{8}) e^{\frac{x}{3}} + \frac{1}{8} e^{- \frac{x}{3}} + \frac{e ^{\frac{x}{3}} x ^{2}}{12} - \frac{1}{8} e^{- \frac{x}{3}} (2 e^{\frac{2 x}{3}} x - 3 e^{\frac{2 x}{3}})

Solution steps

9 y^{''} (x) - y = x e^{\frac{x}{3}}

Solve linear ODE:

y = (- \frac{1}{8} + \frac{5}{8}) e^{\frac{x}{3}} + \frac{1}{8} e^{- \frac{x}{3}} + \frac{e ^{\frac{x}{3}} x ^{2}}{12} - \frac{1}{8} e^{- \frac{x}{3}} (2 e^{\frac{2 x}{3}} x - 3 e^{\frac{2 x}{3}})

y = (- \frac{1}{8} + \frac{5}{8}) e^{\frac{x}{3}} + \frac{1}{8} e^{- \frac{x}{3}} + \frac{e ^{\frac{x}{3}} x ^{2}}{12} - \frac{1}{8} e^{- \frac{x}{3}} (2 e^{\frac{2 x}{3}} x - 3 e^{\frac{2 x}{3}})

\frac{\partial}{\partial y} (\frac{y}{x} + 4 x)

y = e^{x}, y = e^{- x}, x = - 2, x = 1

\int \frac{3 π}{2} cos (\frac{3 π x}{2}) d x

What is the solution for 9y^{''}-y=xe^{x/3},y(0)=1,y^'(0)=0 ?
The solution for 9y^{''}-y=xe^{x/3},y(0)=1,y^'(0)=0 is